The Fast Johnson-Lindenstrauss Transform and Applications

The Fast Johnson-lindenstrauss Transform

While we omit the proof, we remark that it is constructive. Specifically, A is a linear map consisting of random projections onto subspaces of Rd. These projections can be computed by n matrix multiplications, which take time O(nkd). This is fast enough to make the Johnson-Lindenstrauss transform (JLT) a practical and widespread algorithm for dimensionality reduction, which in turn motivates th...

Optimal Fast Johnson-Lindenstrauss Embeddings for Large Data Sets

We introduce a new fast construction of a Johnson-Lindenstrauss matrix based on the composition of the following two embeddings: A fast construction by the second author joint with Ward [1] maps points into a space of lower, but not optimal dimension. Then a subsequent transformation by a dense matrix with independent entries reaches an optimal embedding dimension. As we show in this note, the ...

Some Useful Background for Talk on the Fast Johnson-Lindenstrauss Transform

This writeup includes very basic background material for the talk on the Fast Johnson Lindenstrauss Transform on May 28th, 2007 by the author at the Summer School on Algorithmic Data Analysis in Helsinki, Finland. The talk is about the results in [2], but will be based on a new more recent proof (together with Edo Liberty), using more modern tools, such as probability in Banach spaces and error...

A Survey of Johnson Lindenstrauss Transform - Methods, Extensions and Applications

On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss Transforms

The Johnson-Lindenstrauss lemma is one of the corner stone results in dimensionality reduction. It says that given N , for any set of N vectors X ⊂ Rn, there exists a mapping f : X → Rm such that f(X) preserves all pairwise distances between vectors in X to within (1 ± ε) if m = O(ε lgN). Much effort has gone into developing fast embedding algorithms, with the Fast JohnsonLindenstrauss transfor...